The generaters of positive semigroups.

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David E. Evans and Harald Hanche-Olsen.

University of Oslo.

Abstract: We characterise the infinitesimal generators of nor,m continuous one-parameter semigroups of positive maps

*

on certain ordered spaces, with special reference to C -algebras.

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The purpose of this note is to characterise the infinitesimal generators of norm continuous one-parameter semigroups of positive linear maps on certain ordered vector spaces.

This research was inspired by known results for groups of positive maps in Hilbert spaces with self dual cones [1, Lemma 5o3], and for semigroups of completely positive, and indeed more generally locally completely positive maps on

-¥.·

C -algebras

rs,4].

The study of positive semigroups as

representing dynamical systems in classical and quantum theory has received much attention on recent years. We refer the reader to [ 2, 5, 11] for accounts o.f this.

*

We begin with a result for C -algebras. But first we recall that if

s

is a set of states on a C -algebra A ,

*

then S is said to be full if xE Ah and f(x) > 0 for all f in S , then x > 0 •

invariant if f E S , and x E A

* *

f[x (" )x]jf(x x) E S • That (i)

Moreover S is said to be satisfy f(x*x) ~ 0 , then

is equivalent to (vi) in the following Theorem was first shown in [12], for which we give an alternative proof.

Theorem 1.

Let L be a bounded self-adjoint linear map on a unital C -algebra A •

*

Then the following conditions are equi•

valent:

(i) etL is positive, for all positive t

.

(ii) (A.-L)-1 is positive~ for all large positive ^).

.

(iii) If yEA+, aEA satisfy ya ⁼ 0, then a'L(y)a

*

> 0.

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(iv) For some full, invariant set of states S on A , that yEA+, fE S with f(y) = 0 imply fL(y)

?:

^{0 •}

(v) L(x2 ) + xL(1)x ~ L(x) x + xL(x) , for all self adjoint

x in A •

(vi) L(1) + u L(1)u?. L(u )u + u L(u) , for all unitaries

* * *

u in A •

~

(iv) ^~(iii). Let S be a full, invariant set of states satisfying (iv).

Let yEA+, aE A satisfy ya = 0. Then f(a ya)

*

= 0

for all f in

s.

Hence by (iv), f[a L(y)a] >

*

0 , for all f in

s

since

s

is invariant, and thus

*

a L(y)a > 0 , since . S is full.

(iii)

==;-

(ii). Let A> IlLII _• In order to show that (A- L)- 1

> 0 , it is enough to show that if xE Ah satisfies

(A-L)x ~ 0 , then x ~ 0 • Let x ⁼ x - x ⁺ ^- , where

x+ ,x- E A , and x+x- = 0 • Thus x-JJ(x+)x-

?:

⁰ by (iii) ..

. +

Then we have 0 < x-[(1-L/'-)(x)]x-

= x-x x- -:.i- (L/l.(x)]x-

= -(x-)3 - x- [L/A(x+)]x- + x-[L/A(x-)]x-

Thus·

0 ~.(x-J³~ x-[L/A(x~)]x-, and

so llx-11

³

~ JJ¥1-·IJ

^x-

ff •

Herice x-

=

^{0 ,} as ·I!LI\/). < 1 • (ii) ~(i) etL

=

lim (1 - L/n)-n

n~co

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(i) ~(v) Let L'(x)

=

L(x) - ![L(1)x + xL(1)] • Then etL' > 0 for all t > 0 by the Lie-Trotter formula.

(Note that the map x ~ -![L(1)x + xL(1)] satisfies (i).) Since etL'(1)

=

1 , we have by differentiating Kadison' s Schwarz inequality [ 6], namely

which is valid for all t ^~0 , and all self adjoint x , that

Substituting for L' gives the desired result.

( v) :=;> ( i v) Let y E A+, ^f E A+ , with f ( y)

* =

0 • Then

f(yi z) = 0 = f(zyi) for all z E A , by the Schwarz

1 1 1 A ^~ ^~

inequality. Thus L(y) ⁺ y~L(1)y~ ~ L(y² )y² + y~L(y~)

implies that fL(y) ~ 0 •

To show that (i) is equivalent to (vi), it is enough by a standard transformation to assume that L(1)

=

^{0 •}

(i) ~(vi) Since etL > 0 , and etL(1)

=

1 , for all t > 0 , we have lletL\1

=

1 , Vt > 0 • Thus l!etL(u)!l

S

1 ,

for all unitaries u in A • Hence etL(y*)etL(u) ~ 1 , Vt ^~0 , and so _{L(u )u}

*

+ u L(u)

*

~ 0 , for all unitaries u , by derivation.

(vi) ~(i) This is contained in the proof of [8, Proposition

4]~ as observed in [12].

Not surprisingly, the above theorem is concerned mainly

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with the Jordan structure of the C -algebra. It can be used

*

to simpli~y the proof of the following known result [8,4]

*

_~(C)

*

for the C -structure. Note that if A

= _'

^the

c -

algebra of all n x n matries over 11:

'

^{and L(x)} ⁼

x t -x

"

(where x ^~xt is the transpose mapping) then L satisfies the conditions of Theorem 1 , but not Theorem 2 if n> 2 _•

Theorem 2.

*

Let L be a bounded self-adjoint linear map on a C - algebra A • Then the following conditions are equivalent:

(ii) L(x x)

*

~ L(x )x + x L(x) ,

* *

'tx E A •

Proof.

· Suppose (ii) holds. Adjoin an identity 1 to A and extend L to the enlarged algebra

A

by putting 1(1)

=

^{0 •} Then by Theorem 2 is positive on A ^"' for all positive t • The result follows from [4], or see [8] for a resolvent argument.

It is desirable to remove the initial hypothesis of boundedness of the generator in Theorem 1. We can modify the arguments of [7] to show:

Theorem 3.~

Let L be a self adjoint linear map on a unital C -

*

algebra A, with the following property: if yEA+, fEA+

*

satisfy f(y)

=

0 , then f(Ly) ~ 0 • Then L is bounded, and etL positive for all positive t •

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P.,!OOf.

Note that the map x ~ L(x) -~[L(1)x + xL(1)] satisfies the same conditional positivity property as L • Hence we can assume L(1)

=

0 • We will show that in this case, L is dissipative (in the sense of

[9])

on Ah. i.e.

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A\\xll

:S

II AX - Lx\!,

^'VXE^A_'• ¹ ^>^{0 •}

.·.

In order to show this for some x in Ah, we may assume, (by considering -x if necessary) that there exists a positive f in A* , such that f(x) = llx!l, and llfll = 1 • Then

f(\\xll -x)

=

^{0 ,} and so f[L(\lxl\-x)]

2:

0 by assumption, i.e.

f(Lx) ~ 0 •

Let 1 >

o: ,

then ·1f(x) ~ f().x - Lx)

<

II

f\111 ^{).x -}

Lxl\

i.e. ).1\xll ~

II

fl\11 Ax - Lx\1

This is enough to show that L is bounded. The original proof of this fact used semi-inner products [9]. A more elegant method has been given by Sullivan [11], which we

repeat here for completeness. We show L is closed on Ah • Let fn E Ah , fn ~ 0 , Lfn ~ g • Then for all h in Ah , A. > 0 ,

Letting n -+ c-) , we have

>..llh\1 :: 111-(h-g) + L(h)\1 ,

and then dividing by A , and letting A -+ co , we see

\lh!l

~

1\h-gll

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for all h in Ah • Hence g = 0 • Hence L is bounded on Ah ' and thus on A ^• It follows from Theorem 1 that etL is positive for all t > 0 • (This also follows from ( 1) , which shows that (1 - L/~)-1 is a contraction for ).. >\ILl\ , and thus positive since it preserves the identity).

One of the preceeding techniques is actually derived from a more general setting in certain ordered spaces. For this it is convenient to introduce a definition:

Definition: A cone E+ in a Banach space E is said to have the nearest point property if for any x in E

there exists some y in E+ such that 1\x-y\\

=

dist (x, E+) • In this case E+ is closed.

A closed cone in a Hilbert Epace certainly has the nearest point property. The following Theorem thus improves Connes' characterisation of the generators of groups of

positive maps for Hilbert spaces with self dual cones [1, Lemma 5.3]. The positive cone in any order unit space, in

*

particular, the self-adjoint part of C -algebra, has the nearest point property. Thus the Theorem 4 also generalises the

purely order theoretic part of Theorem 1.

Theorem 4.

Let E be a real Banach space, and E+ a cone in E

with the nearest point property. Then if L is a bounded linear map on E , the following conditions are equivalent:

(i) etL is positive for all positive t •

(ii) (~-L)-¹ is positive for all large positive

A •

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(iii) If satisfy f(x) = 0 , then fL(x) > 0 Proof.

We show (iii) ~(ii). The rest is standard. Let

A > IILI\ ⁰ To show (A.-L)- 1 ::> 0 ' assume xE E and o.-L)xE E+ •

Assuming (iii) holds, we have to show xE E+ • If not, then there exists yE E+ , with \\y-x\\ = dist (x, E+) = d > 0 • A Hahn-Banach theorem applied to the closed convex set E+ , and the interior of the ball S(x,d), with centre x ,

radius d, now provide fEE+ such that f < 0 -)f- on S(x,d) , and f(x) <:: 0 • This implies dl\ fll ~ -f(x) • Since

yE E n S(x,d)

' + f(y)

=

0 , and so by (iii) fL(y)

?::

0 Then 0 ~ f[(1 - L/A.)x] = f(x) - 1/A.fL(x)

= f(x) - 1/A.f(L(x-y)) - 1/AfL(y)

~ - dll fll + 1/ A.ll f\111 Lilli x-y\1

=

dll fll [ 1\LII/t.-1]

< 0 , a contradiction.

The finite dimensional case of the above Theorem was proved in [10]. Following Schneider and Vidyasagar, the

maps satisfying (iii) of Theorem 4 may be called crosspositive.

Intuitively, this means that the orbit {etLx : tEJRJ crosses the boundary of E+ in a positive sense, that is, .:t~~£ E+

(or at least not out of it).

Acknowledgements: We would like to thank E.M. Alfsen and Chr. Skau for stimulating discussions.

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[3] D.E. Evans. Positive linear maps on operator algebras.

Commun. math. Phys. 48, 15-22 (1976).

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[5] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, E.C.G. Sudarshan. Properties of quantum markovian master equations. Preprint 1976.

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Math. 56, 494-503 (1952).

[7] A. Kishimoto. Dissipations and derivations. Commun.

math. Phys. 47, 25-32 (1976).

[8] G. Lindblad. On the generators of quantum dynamical semigroups. Commun. math. Phys. 48, 119-130 (1976).

[9] G. Lumer and R.S. Philips. Dissipative operators in a Banach space. Pac. J. Math. 11, 679-698 (1961).

[10] H. Schneider and M. Vidyasagnr. Cross-positive matrices. Siam J. Numer. Anal. 7, 508-519 (1970).

[11] W.G. Sullivan. Markov processes for random fields.

Comm. T.ubl. Inst. Adv. Studies SerA. 23 (1975).

[12] S.K. Tsui. A note on generators of semigroups.

Preprint 1976.